Prove Proposition (3.21).
Proposition (3.21) states that $a \pmod n$ has an inverse if and only if $\gcd (a, n) = 1$.
($\implies$)
If $x$ is the multiplicative inverse of $a$ modulo $n$ then
$$ ax \equiv 1 \pmod n $$
By Proposition (3.16), for there to be a unique solution to this linear congruence, we must have $\gcd(n,a)=1$.
($\impliedby$)
If $\gcd(a,n)=1$ then by Bezout's Identity there exist integers $x,y$ such that
$$ ax + yn = 1 $$
That is
$$ ax = 1 - yn $$
Or equivalently
$$ ax \equiv 1 \pmod n $$
By showing the implications hold in both directions, we have proven Proposition (3.21).